The FA investigation was centred on Mr Wiseman, who allegedly agreed to deliver grants totalling pounds 3.2m over eight years to the Welsh FA in exchange for votes to support his bid to secure a place on the executive committee of Fifa, world football's governing body.
It is understood that Mr Kelly's signature was on a letter concerning the payments to the Welsh FA, and it is thought his resignation was primarily through shame by association rather than involvement in any wrongdoing.
The news will not only tarnish the image of the English game but could also be damaging to England's bid to stage the 2006 World Cup. Had Mr Wiseman been elected to Fifa's executive committee, his position would have given him access to the people who will ultimately decide the venue for the world's greatest sporting event in 2006. As it is, although the affair will officially have no negative effects on England's chances, it will certainly do nothing to enhance the reputation of the English game in world football.
Mr Kelly's resignation was with immediate effect. Mr Wiseman, who declined to resign, will stay in his position until a special FA meeting next month when he is likely to be sacked.
The FA's vice-chairman, Geoff Thompson, will take the helm at the FA for the immediate future while the FA's director of public affairs, David Davies, will take over as executive director in charge of the daily running of the FA.
At a hastily convened press conference last night, Mr Davies declined to go into specific detail about the affair but read out an official statement.
"The executive committee has this afternoon accepted the resignation of Graham Kelly. They have thanked him for his 10 years of dedicated work and wished him well.
"They have also passed a unanimous vote of no confidence in the chairman, Keith Wiseman. He has declined to resign. The executive will seek endorsement for their action at a special FA council meeting in early January.
"This vote of no confidence was taken unanimously by an executive committee that represents all sections of the game. It was taken with sadness, and only after considerable examination of all the circumstances. It was not taken lightly."
Mr Davies added that yesterday's events will not interfere with the daily work of the FA. "At Lancaster Gate, at Wembley, and everywhere around the country, we have a dedicated staff, all of them football supporters who care passionately about the game."
He added that Glenn Hoddle's position as the England coach would not be affected by the changes at the FA and added that the English FA remained committed to staging the World Cup in 2006.
Commenting on the developments, the Sports Minister, Tony Banks, said in a statement: "It is a matter great sadness to hear of events at the FA. However, it is for the internal workings of the FA to resolve the current situation."
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